0 = n | W| n' = n | Hext| LP = n | Hext| UP 
Direct (phonon mediated)
coupling between the UP and LP is mediated
by the off-diagonal terms in
Eq. and Eq.REF .
For compactness in notation
and to parallel the development in Ref.[1]}
we shall write

w = GLU(b+ b)

and recognize that we will need to average over
the initial vibrational states to obtain our final expressions.
